❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
❊✈♦❧✉çã♦ ♥♦ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s✿
❞❡ ❆rq✉✐♠❡❞❡s ❛ ◆❡✇t♦♥
†♣♦r
❆✉rí❧✐♦ ❞❛ ❙✐❧✈❛ ●✉❡❞❡s
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇♦❝❦❡r ◆❡t♦
s♦❜ ❝♦♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ▼❡✳ ●✐❧♠❛r ❖tá✈✐♦ ❈♦rr❡✐❛
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥✲ t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐✲ ♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❉❡③❡♠❜r♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
♣♦r
❆✉rí❧✐♦ ❞❛ ❙✐❧✈❛ ●✉❡❞❡s
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ●❡♦♠❡tr✐❛ P❧❛♥❛ ❡ ❈á❧❝✉❧♦ ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❇♦❝❦❡r ✲ ❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ▼❡✳ ●✐❧♠❛r ❖tá✈✐♦ ❈♦rr❡✐❛ ✲ ❯❋P❇ ✭❈♦♦r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ❞❡ ❆♥❞r❛❞❡ ❡ ❙✐❧✈❛ ✲ ❯❋P❇
Pr♦❢✳ ❉r✳ ❏♦sé ❱✐❝❡♥t❡ ▼♦r❡✐r❛ ✲ ❯♥✐♣ê
Pr✐♠❡✐r❛♠❡♥t❡ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ♣r♦♣♦r❝✐♦♥❛❞♦ s❛ú❞❡✱ ❢♦rç❛ ❡ ♣❡rs✐s✲ tê♥❝✐❛ ♣❛r❛ ❡♥❢r❡♥t❛r t♦❞♦s ♦s ❞❡s❛✜♦s ❞❡st❛ ❥♦r♥❛❞❛✳
❆ ♠✐♥❤❛ ♠ã❡ ❉♦♥❛ ◆❡♥❡♥✱ q✉❡ s❡♠♣r❡ ♠❡ ❞❡✉ ❝❛r✐♥❤♦✱ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦ ♣❛r❛ ♥✉♥❝❛ ❞❡s✐st✐r✳ ❆♦s ♠❡✉s ✜❧❤♦s ■❛❣♦✱ ●✉st❛✈♦ ❡ ❏ú❧✐❛✱ q✉❡ sã♦ ♦ ♠❡✉ ♥♦rt❡✱ ❡ ✭❝❛❞❛ ✉♠ ❛ s❡✉ ♠♦❞♦✮ ❡♥t❡♥❞❡r❛♠ ❛ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ❞✉r❛♥t❡ ❡st❡ ▼❡str❛❞♦✳
❆♦ ♠❡✉ ✐r♠ã♦ ❆✉ré❧✐♦✱ q✉❡ ♠❡s♠♦ ❞✐st❛♥t❡✱ s❡♠♣r❡ t♦r❝❡✉ ♣♦r ♠✐♠✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❊❞✉❛r❞♦✱ ❙✉❡❧â♥✐♦✱ ❈❤✐❝♦✱ ■r❛③ê✱ ❍❡♥r✐q✉❡ ❡ P❡❞r♦ ❏✉♥✐♦r✱ q✉❡ q✉❛♥❞♦ ♣r❡❝✐s❡✐✱ s❡♠♣r❡ ❧❡✈❛♥t❛r❛♠ ♠✐♥❤❛ ❝❛❜❡ç❛✳
❆ ♠✐♥❤❛ ♥❛♠♦r❛❞❛ ❏✉❧✐❛♥♥❛ ❘♦ss②✱ q✉❡ ❡st❡✈❡ ❝♦♠✐❣♦ ❡♠ t♦❞❛s ❛s ❤♦r❛s✱ ✐♥❝❡♥✲ t✐✈❛♥❞♦ ♦ tér♠✐♥♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆♦s ♣r♦❢❡ss♦r❡s ❇♦❝❦❡r ❡ ●✐❧♠❛r ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ♦r✐❡♥t❛çõ❡s✳
❆ t♦❞♦s ♦s ❝♦♠♣❛♥❤❡✐r♦s ❞♦ P❘❖❋▼❆❚✱ ♣❡❧❛ ❛❝♦❧❤✐❞❛ ❡ ♣❡❧❛ ❢❛♠í❧✐❛ q✉❡ ❢♦r✲ ♠❛♠♦s✳
✶✳✶ ❋✐❣✉r❛s ❞❡ ár❡❛s ❡q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷ ❯♥✐❞❛❞❡ ❞❡ ár❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸ ❘❡tâ♥❣✉❧♦ ❝♦♠ ✷✽ ✉♥✐❞❛❞❡s ❞❡ ár❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✹ ◗✉❛❞r❛❞♦ ❝♦♠ ✶✻ ✉♥✐❞❛❞❡s ❞❡ ár❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✺ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✻ P❛r❛❧❡❧♦❣r❛♠♦ ❆❇❈❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✼ P❛r❛❧❡❧♦❣r❛♠♦ ❳ ❘❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✽ ❚r✐â♥❣✉❧♦ ❳ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✾ ❚r✐â♥❣✉❧♦ ❊q✉✐❧át❡r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✶✵ ❚r✐â♥❣✉❧♦ ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✶✶ ❉❡♠♦♥str❛çã♦ ❞❛ ❋ór♠✉❧❛ ❞❡ ❍❡rã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✶✷ P♦❧í❣♦♥♦ ❈♦♥✈❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✶✸ ▲♦s❛♥❣♦ ❆❇❈❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✶✹ P♦❧í❣♦♥♦s ❘❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✶✺ P♦❧í❣♦♥♦ ■rr❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✷✳✶ ❙❡❣♠❡♥t♦s ❞❡ r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ❉❡♠♦♥str❛çã♦ ❞♦ ▼ét♦❞♦ ❞❛ ❊①❛✉stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ❋✐❣✉r❛ ❞❡ ár❡❛ S ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✷✳✹ P♦❧í❣♦♥♦ ✐♥t❡r✐♦r ❞❡ ár❡❛ ❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✺ ❘❡❧❛çõ❡s ♥❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✻ ❯t✐❧✐③❛çã♦ ❞♦ ▼ét♦❞♦ ❞❛ ❊①❛✉stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✼ ❈♦♥t✐♥✉❛çã♦ ❞♦ ▼ét♦❞♦ ❞❛ ❊①❛✉stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✽ ❈ír❝✉❧♦ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ tr✐â♥❣✉❧♦ ✭❡♠ ár❡❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✾ ✶♦ ❝❛s♦✿ C > T ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✷✳✶✵ ❘❡t✐r❛r tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ❞❡ C−P1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✶✶ ✷♦ ❝❛s♦✿ C < T ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✷✳✶✹ ❙❡❣♠❡♥t♦ ❞❡ ♣❛rá❜♦❧❛ ❞✐✈✐❞✐❞♦ ❡♠ três ♣❛rt❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✶✺ ❙❡❣♠❡♥t♦ ❞❡ ♣❛rá❜♦❧❛ ❞✐✈✐❞✐❞♦ ❡♠ ♥✲✶ ♣❛rt❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✶✻ ◗✉❛❞r❛t✉r❛ ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✶✼ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♣r♦❝❡ss♦ ❞❡ q✉❛❞r❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✸✳✶ ■❞❡✐❛ ❞❡ ❧✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✷ ❯♥✐❝✐❞❛❞❡ ❞♦ ▲✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✸ lim
x→pf(x) = f(p) ❡f ♥ã♦ t❡♠ ❧✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✹ ❋✉♥çã♦ ❝♦♥tí♥✉❛ ① ❋✉♥çã♦ ♥ã♦ ❝♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✺ Pr♦♣r✐❡❞❛❞❡ ❞❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✻ g ♥ã♦ ❝♦♥tí♥✉❛ ❡♠p ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✸✳✼ ❊①❡♠♣❧♦ ❞❡ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✽ ❘❡t❛ ❙❡❝❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✾ ❆♣r♦①✐♠❛çõ❡s ❞❛ r❡t❛ s❡❝❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✶✵ ❘❡❣✐ã♦ ❞♦ ❣rá✜❝♦ ❧✐♠✐t❛❞❛ ♣♦r f(x)♥♦ ✐♥t❡r✈❛❧♦ [a, b] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✸✳✶ ❱❛❧♦r❡s ❞❡ x♣ró①✐♠♦ ❞❡ ✶✱ ❝♦♠ x <1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✸✳✷ ❱❛❧♦r❡s ❞❡ x♣ró①✐♠♦ ❞❡ ✶✱ ❝♦♠ 1< x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✶ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❈♦♥❝❡✐t♦s ❞❡ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s✿ ❚r✐✲
â♥❣✉❧♦s✱ ◗✉❛❞r✐❧át❡r♦s ❡ P♦❧í❣♦♥♦s ❡♠ ●❡r❛❧ ✶✸
✶✳✶ ❯♠❛ ❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷ ❆❧❣✉♥s ❈♦♥❝❡✐t♦s ❞❡ ➪r❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✶ ➪r❡❛ ❞♦ ❘❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✷ ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✸ ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✷✳✹ ➪r❡❛ ❞❡ ✉♠ P♦❧í❣♦♥♦ ◗✉❛❧q✉❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷ ❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❞❡ ❆rq✉✐♠❡❞❡s ✷✼
✷✳✶ ❯♠❛ ❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✶ ❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❡ ❛ ➪r❡❛ ❞♦ ❈ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷✳✷ ❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❡ ❛ ➪r❡❛ ❞❡ ❯♠❛ ❋✐❣✉r❛ P❧❛♥❛ ❉❡❧✐✲
♠✐t❛❞❛ P❡❧♦s ❊✐①♦s ❈♦♦r❞❡♥❛❞♦s ❡ ❯♠ ❙❡❣♠❡♥t♦ P❛r❛❜ó❧✐❝♦ ✳ ✸✼ ✷✳✷✳✸ ❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❡ ❛ ➪r❡❛ ❆❜❛✐①♦ ❞❛ ❋✉♥çã♦ ❈ú❜✐❝❛ ✳ ✳ ✹✶ ✷✳✷✳✹ ❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❡ ❛ ◗✉❛❞r❛t✉r❛ ❞❛ P❛rá❜♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✸ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛s ❆♣❧✐❝❛♥❞♦ ♦ ❈♦♥❝❡✐t♦ ❞❡ ■♥t❡❣r❛❧ ❉❡✜♥✐❞❛ ✺✶ ✸✳✶ ▲✐♠✐t❡ ❞❡ ❯♠❛ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✶✳✶ ◆♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ▲✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✷ ❉❡✜♥✐çã♦ ❡ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✷✳✶ ❯♥✐❝✐❞❛❞❡ ❞♦ ▲✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✸ ❈♦♥t✐♥✉✐❞❛❞❡ ❛ ♣❛rt✐r ❞❛ ❉❡✜♥✐çã♦ ❞❡ ▲✐♠✐t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✹ ❉❡r✐✈❛❜✐❧✐❞❛❞❡ ❞❡ ❯♠❛ ❋✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✹✳✶ ▼♦t✐✈❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✹✳✷ ❉❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✸✳✻ ❉❡✜♥✐çã♦ ❞❡ ■♥t❡❣r❛❧ ❉❡✜♥✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✻✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛ ■♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✻✳✷ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✼ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✼✳✶ ❆♣❧✐❝❛çõ❡s ❞♦ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✹ ❆♣ê♥❞✐❝❡ ✻✾
✹✳✶ ❉❡♥s✐❞❛❞❡ ❞♦s ◆ú♠❡r♦s ❘❛❝✐♦♥❛✐s ❡♠ R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✷ ▲✐♠✐t❡ ❞❡ ❯♠❛ ❙❡q✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✸ ▲✐♠✐t❡s ❡ ❉❡s✐❣✉❛❧❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
❘❡❢❡rê♥❝✐❛s ✼✼
✜❣✉r❛s ♣❧❛♥❛s ❢❡❝❤❛❞❛s✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❛❜♦r❞❛♠♦s ♦ ❝á❧❝✉❧♦ ❞❡st❛s ár❡❛s✳ ◆❛ ♦❜r❛ ❞❡ ❊✉❝❧✐❞❡s✱ ❛ ✐❞❡✐❛ ❞❡ ár❡❛ ❡st❛✈❛ ❛ss♦❝✐❛❞❛ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ✜❣✉r❛s ✭❡q✉✐✈❛❧ê♥❝✐❛✮✱ ♦✉ s❡❥❛✱ ❞✉❛s ✜❣✉r❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s q✉❛♥❞♦ tê♠ ❛ ♠❡s♠❛ ❣r❛♥❞❡③❛ ✭♦✉ ♠❡s♠❛ ár❡❛✮✳ ❖✉tr♦ r❡❝✉rs♦ ✉t✐❧✐③❛❞♦ é ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠❛ ✜❣✉r❛ ❡♠ ♦✉tr❛s ❝✉❥❛s ár❡❛s s❡❥❛♠ ❝♦♥❤❡❝✐❞❛s✳ ❙❡ P é ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛
q✉❡ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st❛ ❡♠ ♥ ♣♦❧í❣♦♥♦sP1✱P2✱· · ·✱Pn✱ t❛✐s q✉❡ ❞♦✐s q✉❛✐sq✉❡r ❞❡❧❡s tê♠ ❡♠ ❝♦♠✉♠ ♥♦ ♠á①✐♠♦ ❛❧❣✉♥s ❧❛❞♦s✱ ❡♥tã♦ ❛ ár❡❛ ❞❡P é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s
♣♦❧í❣♦♥♦sPi✳ ❚♦♠❛♥❞♦ ♦ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✱ ❛♣r❡s❡♥t❛♠♦s ♠♦❞♦s ❞❡ ❞❡t❡r♠✐♥❛r ❛s ár❡❛s ❞❡ ❛❧❣✉♠❛s ✜❣✉r❛s ♣❧❛♥❛s ❢❡❝❤❛❞❛s✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ❛❜♦r❞❛♠♦s ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❞❡ ❆rq✉✐♠❡❞❡s✱ ♦ q✉❛❧ ❛♥t❡❝✐♣♦✉ ♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧✱ ✷✵✵✵ ❛♥♦s ❛♥t❡s ❞❡ t❡r s✐❞♦ ✏❝r✐❛❞♦✑ ♣♦r ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✳ ◆♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♣♦❞❡✲s❡ ❞✐③❡r q✉❡ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❝♦♥s✐st❡ ❡♠ ❜✉s❝❛r ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s ❞❛ ár❡❛ ❛ s❡r ♠❡❞✐❞❛✱ ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦✱ ❛ ♣❛rt✐r ❞❡ ♦✉tr❛s ❥á ❝♦♥❤❡❝✐❞❛s✳ ❆♣❧✐❝❛♠♦s ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ♥❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✱ ♥❛ ár❡❛ ❞❡ s❡❣♠❡♥t♦s ♣❛r❛❜ó❧✐❝♦s ❡ ♥❛ q✉❛❞r❛t✉r❛ ❞❛ ♣❛rá❜♦❧❛✳
❊✱ ❢❡❝❤❛♥❞♦ ❡st❛ ❞✐ss❡rt❛çã♦✱ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❛❜♦r❞❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ❧✐♠✐t❡ ❞❡ ❢✉♥çã♦✱ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ ❞❡r✐✈❛❞❛✱ ♣r✐♠✐t✐✈❛ ❡ ✐♥t❡❣r❛❧ ❞❡✜♥✐❞❛✱ ❝✉❧♠✐♥❛♥❞♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ❡ s✉❛ ❛♣❧✐❝❛çã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❢❡❝❤❛❞❛s✳
❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❈♦♥❝❡✐t♦s ❞❡
➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s✿ ❚r✐â♥❣✉❧♦s✱
◗✉❛❞r✐❧át❡r♦s ❡ P♦❧í❣♦♥♦s ❡♠ ●❡r❛❧
✶✳✶ ❯♠❛ ❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛
❆♦ ♣❡♥s❛r ♥❛ ♦r✐❣❡♠ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ♠❡ ♣❡r❣✉♥t♦ s❡ ❡ss❡ ❝♦♥❝❡✐t♦ t❡r✐❛ s✐❞♦ ❝♦♥str✉í❞♦ ♣❡❧♦ ❤♦♠❡♠ ♦✉ s❡✱ ❛ss✐♠ ❝♦♠♦ ♦ s❡♥s♦ ♥✉♠ér✐❝♦✱ é ✉♠ ❝♦♥❝❡✐t♦ ✐♥❡r❡♥t❡ ❛♦ s❡r ❤✉♠❛♥♦✱ ❛♥t❡s ♠❡s♠♦ q✉❡ ❡❧❡ ♣✉❞❡ss❡ t❡r ❝♦♥s❝✐ê♥❝✐❛ ❞✐ss♦✳
❍á ✐♥❞í❝✐♦s ❤✐stór✐❝♦s ❞❡ q✉❡ ♦❝♦rr❡r❛♠ s♦❝✐❡❞❛❞❡s ❛✈❛♥ç❛❞❛s✱ q✉❡ s❡ ✐♥st❛❧❛r❛♠ ❛♦ ❧♦♥❣♦ ❞♦s r✐♦s ◆✐❧♦✱ ♥♦ ❊❣✐t♦✱ ❚✐❣r❡ ❡ ❊✉❢r❛t❡s✱ ♥❛ ▼❡s♦♣♦tâ♠✐❛✱ ■♥❞♦ ❡ ●❛♥❣❡s✱ ♥♦ ❝❡♥tr♦✲s✉❧ ❞❛ ➪s✐❛ ❡✱ ❍✉❛♥❣ ❤♦ ❡ ❨❛♥❣t③é ♥❛ ➪s✐❛ ♦r✐❡♥t❛❧✳ ❊ss❛s s♦❝✐❡❞❛❞❡s✱ ❝♦♥❤❡❝✐❞❛s ♣♦r s✉❛s ❤❛❜✐❧✐❞❛❞❡s ❡♠ ❡♥❣❡♥❤❛r✐❛✱ ❞r❡♥❛❣❡♠ ❞❡ ♣â♥t❛♥♦s ❡ ✐rr✐❣❛çã♦✱ ❝♦♥str✉ír❛♠ ♦❜r❛s ❞❡ ❞❡❢❡s❛ ❝♦♥tr❛ ✐♥✉♥❞❛çõ❡s✱ ❣r❛♥❞❡s ❡❞✐❢í❝✐♦s ❡ ❡str✉t✉r❛s ♣♦r ♠❡✐♦ ❞❡ ♣r♦❥❡t♦s q✉❡ r❡q✉❡r✐❛♠ ♠✉✐t❛ ❣❡♦♠❡tr✐❛ ♣rát✐❝❛✳
❊ss❛s ❝✐✈✐❧✐③❛çõ❡s ❢♦r❛♠ r❡s♣♦♥sá✈❡✐s ♣❡❧♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠✉✐t❛s t❡❝♥♦❧♦❣✐❛s ❞❛ é♣♦❝❛ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s✱ t❛✐s ❝♦♠♦ ♦ ❝á❧❝✉❧♦ ❞❡ ✉♠ ❝❛❧❡♥❞ár✐♦ ❛❞❡q✉❛❞♦ ❡ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ♣❡s♦s ❡ ♠❡❞✐❞❛s ♣❛r❛ s❡r ✉t✐❧✐③❛❞♦ ♥❛ ❝♦❧❤❡✐t❛✱ ❛r♠❛③❡♥❛♠❡♥t♦ ❡ ❞✐str✐❜✉✐çã♦ ❞❡ ❛❧✐♠❡♥t♦s✳
❆❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ❞♦❝✉♠❡♥t♦s ❤✐stór✐❝♦s ❛t❡st❛♠ ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❣❡♦♠étr✐✲ ❝♦s ❞❛s ❛♥t✐❣❛s ❝✐✈✐❧✐③❛çõ❡s ❛tr❛✈és ❞♦s ♣❛♣✐r♦s ❞❡ ▼♦s❝♦✉ ✭♦✉ ●♦❧❡♥✐s❤❡✈✮ ❡ ❘❤✐♥❞ ✭♦✉ ❆❤♠❡s✮✱ ❞❛t❛❞♦s ❞❡ ✶✽✺✵ ❛✳❈✳ ❡ ✶✻✺✵ ❛✳❈✳✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❆♥á❧✐s❡s ❞❡ss❡s ♣❛♣✐r♦s ❝♦♥st❛t❛r❛♠ q✉❡ ♦s ❡❣í♣❝✐♦s t✐♥❤❛♠ ✈ár✐♦s ❝♦♥❤❡❝✐♠❡♥✲
t♦s ❣❡♦♠étr✐❝♦s ❡ r❡s♦❧✈✐❛♠ ♣r♦❜❧❡♠❛s r❡❧❛❝✐♦♥❛❞♦s à ❣❡♦♠❡tr✐❛✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❊❱❊❙ ✭✶✾✾✺✱ ♣✳✼✺✮✱ ✈✐♥t❡ ❡ s❡✐s ❞♦s ✶✶✵ ♣r♦❜❧❡♠❛s ❞♦s ♣❛♣✐r♦s ❞❡ ▼♦s❝♦✉ ❡ ❘❤✐♥❞ sã♦ ❣❡♦♠étr✐❝♦s✳ ❊ s❡❣✉♥❞♦ ❇❖❨❊❘ ✭✶✾✼✹✱ ♣✳✶✸✮✱ ♦ ♣❛♣✐r♦ ❆❤♠❡s ❝♦♥té♠ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❣❡♦♠étr✐❝♦s✱ ❝♦♠♦ ♦ ♣r♦❜❧❡♠❛ ✺✶✱ q✉❡ ♠♦str❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s ❛tr❛✈és ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ♠❡t❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❛ ❜❛s❡ ♣❡❧❛ ♠❡✲ ❞✐❞❛ ❞❛ ❛❧t✉r❛✱ ❡ ♦ ♣r♦❜❧❡♠❛ ✺✷ q✉❡ tr❛t❛ ❞❛ ár❡❛ ❞♦ tr❛♣é③✐♦ ✐sós❝❡❧❡s ❞❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✳
❍á t❛♠❜é♠ ✐♥❞í❝✐♦s ❞❡ q✉❡ ❡❣í♣❝✐♦s ❡ ❜❛❜✐❧ô♥✐♦s ❞✐s♣✉♥❤❛♠ ❞❡ ♠ét♦❞♦s ❡✜❝✐❡♥✲ t❡s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❡ ❝♦♥❤❡❝✐❛♠ r❡❣r❛s ❣❡r❛✐s ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ tr✐â♥❣✉❧♦s✱ r❡tâ♥❣✉❧♦s ❡ tr❛♣é③✐♦s✱ ❡ ❛s ✉t✐❧✐③❛✈❛♠ ♣❛r❛ ❝❛❧❝✉❧❛r✱ ❞❡ ❢♦r♠❛ ❛♣r♦①✐✲ ♠❛❞❛✱ ❛s ♠❡❞✐❞❛s ❞♦s t❡rr❡♥♦s ❝✉❧t✐✈❛❞♦s✱ ♠❡s♠♦ q✉❛♥❞♦ t✐♥❤❛♠ ❛ ❢♦r♠❛ ❞❡ ✜❣✉r❛s ♠❛✐s ❝♦♠♣❧❡①❛s✳ ❊♠ ❣❡r❛❧ ❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ✉t✐❧✐③❛❞❛ ❡r❛ ✉♠ q✉❛❞r❛❞♦✱ ♠❛s ❡♠ ❛❧❣✉♠❛s s✐t✉❛çõ❡s ❛ ❡str❛té❣✐❛ ✉t✐❧✐③❛❞❛ ❡r❛ ❞❡❝♦♠♣♦r ❛ s✉♣❡r❢í❝✐❡ ❡♠ tr✐â♥❣✉❧♦s ♦✉ r❡tâ♥❣✉❧♦s ❡ ❝❛❧❝✉❧❛r ❛ s✉❛ ár❡❛ ❝♦♠♦ ❛ s♦♠❛ ❞❛s ár❡❛s ❞❛s r❡❣✐õ❡s r❡s✉❧t❛♥t❡s ❞❡st❛ ❞❡❝♦♠♣♦s✐çã♦✳
◆ã♦ s❡ s❛❜❡ ❛♦ ❝❡rt♦ ♣♦rq✉❡ ♦ q✉❛❞r❛❞♦ ❢♦✐ ❡s❝♦❧❤✐❞♦ ♣❛r❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✱ t❛❧✈❡③ ♣♦r s❡r ❛ ✜❣✉r❛ ♣❧❛♥❛ ♠❛✐s s✐♠♣❧❡s✳ ❉❡♥tr❡ ❛s ♠✉✐t❛s ✈❡rsõ❡s ❛♣r❡s❡♥t❛❞❛s✱ ✉♠❛ ❞❡❧❛s é q✉❡ ❛ ❡s❝♦❧❤❛ ❢♦✐ ✐♥s♣✐r❛❞❛ ♣❡❧❛ ♠❛♥❡✐r❛ ❞❡ s❡ t❡❝❡r ✉♠❛ ❝❡st❛✱ ❛rt❡ q✉❡ ♣r❡❝❡❞❡✉ ❛ ✜❛çã♦✳ ❖✉tr❛✱ q✉❡ ❢♦✐ ♦ r❡s✉❧t❛❞♦ ❞♦ ✉s♦ ❞❡ ❧❛❞r✐❧❤♦s ❞❡ ♠♦s❛✐❝♦s ❤✐♥✲ ❞✉s ❡ ❝❤✐♥❡s❡s ♦✉ s✉❣❡r✐❞❛s ♣❡❧♦s ♣❛❞rõ❡s q✉❛❞r✐❝✉❧❛❞♦s q✉❡ ❞❡❝♦r❛✈❛♠ ❛ ❝❡râ♠✐❝❛ ♣r♦❞✉③✐❞❛ ♣❡❧♦s ❜❛❜✐❧ô♥✐♦s✳
◆❛ ●ré❝✐❛ ❛♥t✐❣❛✱ ♣♦r ✈♦❧t❛ ❞❡ ✸✵✵ ❛✳❈✳ ✱ ♦ ❣❡ô♠❡tr❛ ❣r❡❣♦ ❊✉❝❧✐❞❡s ♣r♦❞✉③✐❛ s✉❛ ♦❜r❛ ♣r✐♠❛ ✐♥t✐t✉❧❛❞❛ ❖s ❊❧❡♠❡♥t♦s✱ q✉❡ r❡✉♥✐✉ ❞❡ ♠♦❞♦ s✐st❡♠❛t✐③❛❞♦ ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ s❡✉s ♣r❡❝✉rs♦r❡s✳ ❆ ♠❛✐♦r ♣❛rt❡ ❞♦ ❝♦♥t❡ú❞♦ ❞❛ ♦❜r❛ s❡ r❡❢❡r❡ à ❣❡♦♠❡tr✐❛✱ ❡♥tr❡t❛♥t♦ t❛♠❜é♠ ❝♦♥t❡♠♣❧❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❡ á❧❣❡❜r❛ ❡❧❡♠❡♥t❛r ♦✉ ❣❡♦♠étr✐❝❛✳ ❊ss❛ ♦❜r❛ t❡♠ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ ♥❛ ❢♦r♠❛ ❝♦♠♦ tr❛t❛♠♦s ❛ ❣❡♦♠❡tr✐❛ ♥♦s ❝✉rrí❝✉❧♦s ❡s❝♦❧❛r❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳
◆❛ ♦❜r❛ ❞❡ ❊✉❝❧✐❞❡s✱ ❛ ✐❞❡✐❛ ❞❡ ár❡❛ ❡stá ❛ss♦❝✐❛❞❛ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ✜❣✉r❛s ✭❡q✉✐✈❛❧ê♥❝✐❛✮✳ ■st♦ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞♦ q✉❛♥❞♦ ❡♥✉♥❝✐❛ q✉❡ tr✐â♥❣✉❧♦s ❝♦♠ ❜❛s❡s ✐❣✉❛✐s✱ s✐t✉❛❞♦s ❡♥tr❡ ❛s ♠❡s♠❛s ♣❛r❛❧❡❧❛s sã♦ ✜❣✉r❛s ✐❣✉❛✐s ✭❡q✉✐✈❛❧❡♥t❡s✮✱ ❡ q✉❡ ♣❛r❛❧❡❧♦❣r❛♠♦s ❝♦♠ ❜❛s❡s ✐❣✉❛✐s s✐t✉❛❞❛s ❡♥tr❡ ❛s ♠❡s♠❛s ♣❛r❛❧❡❧❛s t❛♠❜é♠ sã♦ ✜❣✉r❛s ✐❣✉❛✐s✳ ❖✉ s❡❥❛✱ ❞✉❛s ✜❣✉r❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s q✉❛♥❞♦ tê♠ ❛ ♠❡s♠❛ ❣r❛♥❞❡③❛ ✭♦✉ ♠❡s♠❛ ár❡❛✮✳
❋✐❣✉r❛ ✶✳✶✿ ❋✐❣✉r❛s ❞❡ ár❡❛s ❡q✉✐✈❛❧❡♥t❡s
✶✳✷ ❆❧❣✉♥s ❈♦♥❝❡✐t♦s ❞❡ ➪r❡❛
• ▼❡❞✐r ✉♠❛ ❣r❛♥❞❡③❛ ✭s✉♣❡r❢í❝✐❡✮ s✐❣♥✐✜❝❛ ❝♦♠♣❛rá✲❧❛ ❝♦♠ ♦✉tr❛ ❞❡ ♠❡s♠❛
❡s♣❡❝✐❡✱ t♦♠❛❞❛ ❝♦♠♦ ✉♥✐❞❛❞❡✳ ❖ r❡s✉❧t❛❞♦ ❞❡ss❛ ❝♦♠♣❛r❛çã♦ é ✉♠ ♥ú♠❡r♦✱ ❡ss❡ ♥ú♠❡r♦ é ❛ ♠❡❞✐❞❛✳ ✭❊❞✉❛r❞♦ ❲❛❣♥❡r✮
• ➪r❡❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❧✐♠✐t❛❞❛ é ♦ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ ❛ss♦❝✐❛❞♦ à s✉♣❡r❢í❝✐❡
❞❡ t❛❧ ❢♦r♠❛ q✉❡✿
✐✳ ❆s s✉♣❡r❢í❝✐❡s ❡q✉✐✈❛❧❡♥t❡s ❡stã♦ ❛ss♦❝✐❛❞❛s ❛ ár❡❛s ✐❣✉❛✐s ❡ r❡❝✐♣r♦❝❛✲ ♠❡♥t❡✳
✐✐✳ ❆ ✉♠❛ s♦♠❛ ❞❡ s✉♣❡r❢í❝✐❡s ❡stá ❛ss♦❝✐❛❞❛ ✉♠❛ ár❡❛ q✉❡ é ❛ s♦♠❛ ❞❛s ár❡❛s ❞❛s s✉♣❡r❢í❝✐❡s ♣❛r❝❡❧❛s✳
✐✐✐✳ ❙❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡stá ❝♦♥t✐❞❛ ❡♠ ♦✉tr❛✱ ❡♥tã♦ s✉❛ ár❡❛ é ♠❡♥♦r ✭♦✉ ✐❣✉❛❧✮ ❞♦ q✉❡ ❛ ár❡❛ ❞❛ ♦✉tr❛✳
✭●❊▲❙❖◆ ■❊❩❩■✮
◆♦s ❛t✉❛✐s ♣❛râ♠❡tr♦s ❡s❝♦❧❛r❡s ❞♦ ❇r❛s✐❧✱ ♣r❡✈❛❧❡❝❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ❝♦♠♦ ❣r❛♥❞❡③❛✱ ❛ss✐♠✱ ✈❡❥♦ ✉♠ ❜♦♠ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ❝♦♠♦ ♦ s❛❜❡r ♠❛t❡♠át✐❝♦ q✉❡ ♣❡r✲ ♠✐t❡ ❝♦♠♣❛r❛r ❡ ♠❡❞✐r ✉♠❛ s✉♣❡r❢í❝✐❡✳ ❋❛❧❛♥❞♦ ❡♠ s✉♣❡r❢í❝✐❡✱ ✈♦✉ ♠❡ r❡❢❡r✐r ❛ ✉♠❛ ♣♦rçã♦ ❞♦ ♣❧❛♥♦ ❧✐♠✐t❛❞❛ ♣♦r ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ ❢❡❝❤❛❞❛✳ ▼❡❞✐r ✉♠❛ s✉♣❡r❢í❝✐❡ s✐❣✲ ♥✐✜❝❛ ♦❜t❡r ✉♠ ♥ú♠❡r♦ q✉❡ r❡♣r❡s❡♥t❡ ❛ ♣♦rçã♦ ❞♦ ♣❧❛♥♦ ♦❝✉♣❛❞❛ ♣♦r ❡ss❛ r❡❣✐ã♦✱ ❝♦♠♣❛r❛❞❛ ❝♦♠ ✉♠❛ ♦✉tr❛ ✜①❛❞❛✳ ❊ss❛ ♠❡❞✐❞❛ é ❝❤❛♠❛❞❛ ❞❡ ár❡❛✳
❆ss✐♠✱ ♣❛r❛ ♠❡❞✐r ❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠❛ r❡❣✐ã♦ é ♥❡❝❡ssár✐♦ ✉♠❛ ♦✉tr❛ s✉♣❡r❢í❝✐❡ ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❡ ✈❡r✐✜❝❛r q✉❛♥t❛s ✈❡③❡s ❡ss❛ ✉♥✐❞❛❞❡ ❝❛❜❡ ❞❡♥tr♦ ❞❡ss❛ r❡❣✐ã♦ ❛ s❡r ♠❡❞✐❞❛✳ ❊♠ ❣❡r❛❧✱ t♦♠❛✲s❡ ✉♠ q✉❛❞r❛❞♦ ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❡ ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s ♦❜t✐❞♦ é ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ♠❡❞✐❞❛✳ ❖✉tr♦ r❡❝✉rs♦ ✉t✐❧✐③❛❞♦ é ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠❛ ✜❣✉r❛ ❡♠ ♦✉tr❛s ❝✉❥❛s ár❡❛s s❡❥❛♠ ❝♦♥❤❡❝✐❞❛s✳
❊ss❛s ❡str❛té❣✐❛s tê♠ s✉❛s ♦r✐❣❡♥s ♥♦s ❛♥t✐❣♦s ♠♦❞♦s ❞❡ ♠❡❞✐r✱ ♠❛s ♣♦❞❡♠ s❡r ❡s❝r✐t❛s ❡♠ ❧✐♥❣✉❛❣❡♠ ❢♦r♠❛❧✱ ❝♦♠♦ ❡♠ ▲■▼❆ ✭✶✾✾✶✱ ♣✳✷✶✮✱ ❛tr❛✈és ❞❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
❙❡❥❛ P ✉♠ ♣♦❧í❣♦♥♦ ❞♦ ♣❧❛♥♦✳ ❆ ❝❛❞❛ ♣♦❧í❣♦♥♦ P ♣♦❞❡✲s❡ ❛ss♦❝✐❛r ✉♠ ♥ú♠❡r♦
r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦✱ ❝❤❛♠❛❞♦ ❞❡ ár❡❛ ❞❡ P✱ ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
• P♦❧í❣♦♥♦s ❝♦♥❣r✉❡♥t❡s tê♠ ár❡❛s ✐❣✉❛✐s✳
• ❙❡P é ✉♠ q✉❛❞r❛❞♦ ❝♦♠ ❧❛❞♦ ✉♥✐tár✐♦✱ ❡♥tã♦ ❛ ár❡❛ ❞❡ P é ✐❣✉❛❧ ❛ ✶✳
• ❙❡P é ✉♠❛ ✜❣✉r❛ ♣❧❛♥❛ q✉❡ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st❛ ❡♠n ♣♦❧í❣♦♥♦s P1✱P2✱ · · ·✱
Pn✱ t❛✐s q✉❡ ❞♦✐s q✉❛✐sq✉❡r ❞❡❧❡s tê♠ ❡♠ ❝♦♠✉♠ ♥♦ ♠á①✐♠♦ ❛❧❣✉♥s ❧❛❞♦s✱ ❡♥tã♦ ❛ ár❡❛ ❞❡P é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s ♣♦❧í❣♦♥♦sPi✳
❆ ♣❛rt✐r ❞❡ss❛s ✐❞❡✐❛s ✈❛♠♦s ❛♣r❡s❡♥t❛r ♠♦❞♦s ❞❡ ❞❡t❡r♠✐♥❛r ❛s ár❡❛s ❞❡ ❛❧❣✉♠❛s ✜❣✉r❛s ♣❧❛♥❛s✳ P❛r❛ t❛♥t♦✱ ❛ss✉♠✐r❡♠♦s ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ✉♠ q✉❛❞r❛❞♦ ❝✉❥♦ ❧❛❞♦ ♠❡❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✭✉✳❝✳✮✱ q✉❡ s❡rá ❝❤❛♠❛❞♦ ❞❡ q✉❛❞r❛❞♦ ✉♥✐tár✐♦✳ ❆ss✐♠✱ ❛ ár❡❛ ❞❡ss❡ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ s❡rá ✐❣✉❛❧ ❛ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ ✭✉✳❛✳✮✳
❋✐❣✉r❛ ✶✳✷✿ ❯♥✐❞❛❞❡ ❞❡ ár❡❛
✶✳✷✳✶ ➪r❡❛ ❞♦ ❘❡tâ♥❣✉❧♦
▲❡♠❛ ✶✿ ❙❡❥❛ ♥ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ♦ q✉❛❞r❛❞♦Q ❞❡ ❧❛❞♦ 1
n✱ t❡♠ ár❡❛
✐❣✉❛❧ ❛ 1
n2✳ ❉❡❝♦♠♣♦♥❞♦ ♦ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ ♣♦r ♠❡✐♦ ❞❡ ♣❛r❛❧❡❧❛s ❛♦s s❡✉s ❧❛❞♦s✱ ❡♠ n2 q✉❛❞r❛❞♦s ❥✉st❛♣♦st♦s ❛ Q✱ ❡st❡s n2 q✉❛❞r❛❞♦s ❝♦♥❣r✉❡♥t❡s ❛ Q❢♦r♠❛♠ ✉♠ q✉❛❞r❛❞♦ ❞❡ ár❡❛ ✶✳ ❉❛í✱ ❛ ár❡❛ ❞❡ Q ❞❡✈❡ s❛t✐s❢❛③❡r ❛ ❝♦♥❞✐çã♦ n2 ×A(Q) = 1✱ ❡♥tã♦ s❡❣✉❡✲s❡ q✉❡A(Q) = 1
n2✳
❖ r❡tâ♥❣✉❧♦ é ♦ q✉❛❞r✐❧át❡r♦ q✉❡ ♣♦ss✉✐ q✉❛tr♦ â♥❣✉❧♦s r❡t♦s ❡ ❧❛❞♦s ♦♣♦st♦s ♣❛r❛❧❡❧♦s✳
❚❡♦r❡♠❛ ✶ ❉❛❞♦ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s a ❡ b r❛❝✐♦♥❛✐s ♦✉ ♥ã♦✱ s✉❛ ár❡❛ s❡rá S =ab✳
❋✐❣✉r❛ ✶✳✸✿ ❘❡tâ♥❣✉❧♦ ❝♦♠ ✷✽ ✉♥✐❞❛❞❡s ❞❡ ár❡❛
❉❡♠♦♥str❛çã♦✿
P❛r❛ ✈❡r ✐ss♦✱ ❞✐✈✐❞✐r❡♠♦s ❛ ♣r♦✈❛ ❡♠ ❞✉❛s ♣❛rt❡s✳ ❆ ♣r✐♠❡✐r❛ é s✉♣♦r q✉❡ a ❡b
sã♦ r❛❝✐♦♥❛✐s ♣♦s✐t✐✈♦s✱ ✐st♦ é✿
a= m1
n1 ❡
b = m2
n2 ✭✶✳✶✮
♦♥❞❡ m1✱ n1✱m2 ❡n2 sã♦ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✳ ❉❡ ✶✳✶✱ s❡❣✉❡✲s❡ q✉❡✿
a = m1
n1
= m1n2
n1n2 ❡
b = m2
n2
= m2n1
n22n1✳ ✭✶✳✷✮
❆ ❣r❛♥❞❡③❛ 1
m =
1
n1n2 é ❛ ♠❡❞✐❞❛ ❝♦♠✉♠ ❞♦s ❞♦✐s ❧❛❞♦s✳ ❆ss✐♠✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛s ❡①♣r❡ssõ❡s ❞❛❞❛s ❡♠ ✶✳✷ ♣♦r
a= m1n2
m ❡ b= m2n1
m ✳ ✭✶✳✸✮
P❛r❛ ❝♦♥❝❧✉✐r ❡st❡ ❝❛s♦✱ s✉❜❞✐✈✐❞✐♠♦s ♦ r❡tâ♥❣✉❧♦ ❡♠ ♣❡q✉❡♥♦s q✉❛❞r❛❞♦s ❞❡ ❧❛❞♦ 1
m✱ ❝✉❥❛ ár❡❛ ❞❡ ❝❛❞❛ q✉❛❞r❛❞♦ é
1
m2 ✳ ❖❜s❡r✈❡ q✉❡ ♦ ♥ú♠❡r♦ t♦t❛❧ N ❞❡ss❡s q✉❛❞r❛❞♦s é ❞❛❞♦ ♣♦r
N = a1
m
b
1
m
=abm2 =m
1n2m2n1.
❆ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❢♦✐ ♦❜t✐❞❛ ✉s❛♥❞♦ ✭✶✳✸✮✳ ▲♦❣♦ ❛ ár❡❛ S ❞♦ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧
❛♦ ♥ú♠❡r♦ t♦t❛❧ ❞❡ ✏q✉❛❞r❛❞✐♥❤♦s✑ ✈❡③❡s ❛ ár❡❛ ❞❡ ❝❛❞❛ ✏q✉❛❞r❛❞✐♥❤♦✑✱ ♦✉ s❡❥❛
S =N 1
m2 =m1n2m2n1
1
m2 =
m1n2m2n1
n2 1n22
= m1
n1
m2
n2
=ab.
P❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡a❡bsã♦ ✐rr❛❝✐♦♥❛✐s✱ ✉s❛♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ ❡①✐st❡♠ s❡q✉ê♥❝✐❛s
❞❡ r❛❝✐♦♥❛✐s (xn) ❡(yn) t❛✐s q✉❡
lim
n→∞xn=a ❡ nlim→∞yn =b✳ ✭✶✳✹✮ ❙❡❥❛ Sn ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s xn ❡ yn✳ ❙❡♥❞♦ ❡ss❡s ❧❛❞♦s r❛❝✐♦♥❛✐s✱ ♣❡❧♦ ✐t❡♠ ❛♥t❡r✐♦r✱ t❡♠♦sSn =xnyn✳ ❉❡✜♥✐♥❞♦✿
S = lim
n→∞Sn
♠♦str❛r❡♠♦s q✉❡ S = ab ❝♦♠ a ❡ b ✐rr❛❝✐♦♥❛✐s✳ ❉❡ ❢❛t♦✱ ✉s❛♥❞♦ ❛s ❡①♣r❡ssõ❡s
✭✶✳✹✮✱ t❡♠♦s✿
S = lim
n→∞Sn = limn→∞xnyn= limn→∞xnnlim→∞yn=ab.
❈♦r♦❧ár✐♦ ✶✿ ❈♦♠♦ ♦ q✉❛❞r❛❞♦ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠ r❡tâ♥❣✉❧♦ q✉❡ ♣♦ss✉✐ t♦❞♦s ♦s ❧❛❞♦s ✐❣✉❛✐s✱ s✉❛ ár❡❛ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦✳
❋✐❣✉r❛ ✶✳✹✿ ◗✉❛❞r❛❞♦ ❝♦♠ ✶✻ ✉♥✐❞❛❞❡s ❞❡ ár❡❛
❆ss✐♠✱ ♣♦❞❡♠♦s ❡①♣r❡ss❛r ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ Q✱ ❝✉❥♦ ❧❛❞♦ ♠❡❞❡ a ❞❛ s❡❣✉✐♥t❡
❢♦r♠❛✿
➪r❡❛ ❞♦ q✉❛❞r❛❞♦ Q=a2✳
✶✳✷✳✷ ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦
❖ ♣❛r❛❧❡❧♦❣r❛♠♦ é ♦ q✉❛❞r✐❧át❡r♦ q✉❡ t❡♠ ❧❛❞♦s ♦♣♦st♦s ♣❛r❛❧❡❧♦s✱ ❛ss✐♠ ❝♦♠♦ ♦ r❡tâ♥❣✉❧♦✳ ❊♥tr❡t❛♥t♦✱ ❛♦ t❡♥t❛r♠♦s s♦❜r❡♣♦r q✉❛❞r❛❞♦s ✉♥✐tár✐♦s ♣❛r❛ ♦❜t❡r s✉❛ ár❡❛✱ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❛❧❣✉♠❛s ❧✐♠✐t❛çõ❡s✱ ♣♦✐s ♦s s❡✉s â♥❣✉❧♦s ✐♥t❡r♥♦s ♣♦❞❡♠ ♥ã♦ s❡r r❡t♦s✳
P❛r❛ ❞❡✜♥✐r ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ r❡❝♦rr❡♠♦s à ❞❡❝♦♠♣♦s✐çã♦✱ ❞❡ ♠♦❞♦ ❛ ❝♦♠♣❛rá✲❧❛ ❝♦♠ ♦✉tr❛ ár❡❛ ❥á ❝♦♥❤❡❝✐❞❛✱ ♥♦ ❝❛s♦ ♦ r❡tâ♥❣✉❧♦✳
❋✐❣✉r❛ ✶✳✺✿ P❛r❛❧❡❧♦❣r❛♠♦
❊♠ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ q✉❛♥❞♦ s❡ t♦♠❛ ✉♠ ❞❡ s❡✉s ❧❛❞♦s ❝♦♠♦ ❜❛s❡✱ ❝❤❛♠❛✲s❡ ❛❧t✉r❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ à ❞✐stâ♥❝✐❛ ❡♥tr❡ ❡ss❛ ❜❛s❡ ❡ ♦ s❡✉ ❧❛❞♦ ♦♣♦st♦✳
◆♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD✱ t♦♠❛♥❞♦✲s❡AB ❝♦♠♦ ❜❛s❡ ❞❡ ♠❡❞✐❞❛ a✱ ♦ s❡❣♠❡♥t♦ EF✱ ❞❡ ♠❡❞✐❞❛b✱ r❡♣r❡s❡♥t❛ ❛ s✉❛ ❛❧t✉r❛ ✭❋✐❣✉r❛ ✶✳✻✮✳
❋✐❣✉r❛ ✶✳✻✿ P❛r❛❧❡❧♦❣r❛♠♦ ❆❇❈❉
P❛r❛ ♦❜t❡r ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡❢❡t✉❛♠♦s ✉♠ ❝♦rt❡ ❛♦ ❧♦♥❣♦ ❞❡ s✉❛ ❛❧t✉r❛
EF✱ ❡ ❡♠ s❡❣✉✐❞❛ r❡❝♦♠♣♦♠♦s ❛s ♣❛rt❡s ❞❡ ♠♦❞♦ ❛ ❢♦r♠❛r ✉♠ r❡tâ♥❣✉❧♦✳
❋✐❣✉r❛ ✶✳✼✿ P❛r❛❧❡❧♦❣r❛♠♦ ❳ ❘❡tâ♥❣✉❧♦
❖ r❡tâ♥❣✉❧♦ ❢♦r♠❛❞♦ t❡♠ ❛s ❞✐♠❡♥sõ❡s a ❡ b✱ ❡ s✉❛ ár❡❛ é ❞❛❞❛ ♣♦r ab✳ ❆ss✐♠✱
♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♣r♦❞✉t♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❞❡ s✉❛s ❜❛s❡s ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡✳
✶✳✷✳✸ ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦
❆ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❞✐r❡t❛♠❡♥t❡ ❛ ♣❛rt✐r ❞❛ ár❡❛ ❞♦ ♣❛r❛❧❡✲ ❧♦❣r❛♠♦✱ ✈✐st♦ q✉❡ tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s tê♠ ♠❡s♠❛ ár❡❛✱ ❡ t♦❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st♦ ❡♠ ❞♦✐s tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s✳
◆♦ tr✐â♥❣✉❧♦ ABC ❡ ♥♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD✱ ❛ ❜❛s❡ AB t❡♠ ♠❡❞✐❞❛ b ❡ a
❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ CE t❡♠ ♠❡❞✐❞❛ a✳
❋✐❣✉r❛ ✶✳✽✿ ❚r✐â♥❣✉❧♦ ❳ P❛r❛❧❡❧♦❣r❛♠♦
❆ ár❡❛ ❞❡ ABCD =ab✱ ❡ ♦s tr✐â♥❣✉❧♦s ABC ❡BCD sã♦ ❝♦♥❣r✉❡♥t❡s✱ ♣♦rt❛♥t♦
❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC é ❛ ♠❡t❛❞❡ ❞❛ ár❡❛ ❞❡ ABCD✳ ❖✉ ❛✐♥❞❛✱ ❛ ár❡❛ ❞❡ ✉♠
tr✐â♥❣✉❧♦ é ❛ ♠❡t❛❞❡ ❞♦ ♣r♦❞✉t♦ ❞❛ ♠❡❞✐❞❛ ❞❡ ✉♠❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛ ❝♦rr❡s♣♦♥❞❡♥t❡✳ ➪r❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC = 1
2ab✳
P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❛ ár❡❛ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠❛✐s ❛♠✐❣á✈❡❧✱ q✉❛♥❞♦ ♦ tr✐â♥❣✉❧♦ t❡♠ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❡s♣❡❝✐❛❧✱ ❡①❡♠♣❧♦s✿
➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ ❊q✉✐❧át❡r♦
❋✐❣✉r❛ ✶✳✾✿ ❚r✐â♥❣✉❧♦ ❊q✉✐❧át❡r♦
❈♦♥s✐❞❡r❛♥❞♦ ♦ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ABC ❞❡ ❧❛❞♦ ❧ ❡ ❛❧t✉r❛ h✱ ✈❛♠♦s ❝❛❧❝✉❧❛r
♣r✐♠❡✐r♦ ♦ ✈❛❧♦r ❞❡ h✱ ❛♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♥♦ tr✐â♥❣✉❧♦ ADB✿
l2 =h2+
l
2
2
l2 =h2+l
2
4 4l2 = 4h2+l2
4h2 = 3l2
h2 = 3l
2
4
h= l 2
√
3
❈♦♠♦ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ é ❞❛❞❛ ♣♦r A = 12lh✱ s❡❣✉❡✲s❡ q✉❡✿
A = 1 2l
l
2
√
3
A = l
2
4
√
3
❋ór♠✉❧❛ ❞❡ ❍❡rã♦
❊ss❛ ❢ór♠✉❧❛ é ❜❛st❛♥t❡ ✉s❛❞❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❝♦♥❤❡❝❡♥❞♦✲ s❡ ❛ ♠❡❞✐❞❛ ❞♦s s❡✉s ❧❛❞♦s✳
❈♦♥s✐❞❡r❡✲s❡ ♦ s❡❣✉✐♥t❡ tr✐â♥❣✉❧♦ ❞❡ ✈ért✐❝❡s A✱B ❡C✱ â♥❣✉❧♦s ✐♥t❡r♥♦s ✭A✱B ❡ C✮ ❡ ❧❛❞♦s a✱ b ❡ c✳
❋✐❣✉r❛ ✶✳✶✵✿ ❚r✐â♥❣✉❧♦ ❆❇❈
❆ ❢ór♠✉❧❛ ❞❡ ❍❡rã♦ ❞❛ ár❡❛ S ❞❡ ✉♠ tr✐â♥❣✉❧♦ é✿
S =pp(p−a)(p−b)(p−c)
❡♠ q✉❡ p é ♦ s❡♠✐✲♣❡rí♠❡tr♦ ❞♦ tr✐â♥❣✉❧♦✿
p= a+b+c
2 .
❉❡♠♦♥str❛çã♦✿
❋✐❣✉r❛ ✶✳✶✶✿ ❉❡♠♦♥str❛çã♦ ❞❛ ❋ór♠✉❧❛ ❞❡ ❍❡rã♦
❙❡❥❛CP =h ❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ABC tr❛ç❛❞❛ s♦❜r❡ ♦ ❧❛❞♦ AB ✭❋✐❣✉r❛ ✶✳✶✶✮✳
❖ ♣♦♥t♦ P✱ ♣r♦❥❡çã♦ ❞♦ ✈ért✐❝❡ C s♦❜r❡ ♦ ❧❛❞♦ AB✱ ❞✐✈✐❞❡✲♦ ❡♠ ❞♦✐s s❡❣♠❡♥t♦s AP =r ❡ P B =s✱ t❛✐s q✉❡ AB = c= r+s✳ ❖ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❛♣❧✐❝❛❞♦ ❛♦
tr✐â♥❣✉❧♦ ACP tr❛❞✉③✲s❡ ❡♠✿
b2 =r2+h2 ✭✶✳✺✮
❡ ❛♣❧✐❝❛❞♦ ❛♦ tr✐â♥❣✉❧♦ BCP ❡♠
h2 =a2−s2 ✭✶✳✻✮
❞❡ r=c−s✱ ♦❜t❡♠♦s
r2 =c2−2cs+s2 ✭✶✳✼✮
❙✉❜st✐t✉✐♥❞♦ ✭✶✳✻✮ ❡ ✭✶✳✼✮ ❡♠ ✭✶✳✺✮ ✈❡♠✿
b2 =c2+a2−2cs ✭✶✳✽✮
❞♦♥❞❡
s= 1 2c(c
2+a2
−b2) ✭✶✳✾✮
❊❧✐♠✐♥❛♥❞♦ s ❡♠ ✭✶✳✻✮✱ t❡♠✲s❡ s✉❝❡ss✐✈❛♠❡♥t❡
h2 =a2−
c2+a2−b2
2c
2
= 4c
2a2 −(c2+a2−b2)2
4c2
= 1
4c2[2ac−(c
2+a2−b2)][2ac+ (c2 +a2−b2)]
= 1 4c2[b
2−(a2 −2ac+c2)][(a2+ 2ac+c2)−b2]
= 1 4c2[b
2−(a−c)2][(a+c)2−b2]
= 1
4c2[b−(a−c)][b+ (a−c)][a+c−b][a+c+b]
= 1
s❡♥❞♦ p= a+b+c
2 ⇒a+b+c= 2p❀ a+c−b = 2−2b= 2(p−b)❀a+b−c= 2p−2c= 2(p−c) ❡ b+c−a= 2p−2a = 2(p−a)⇒
h2 = 1
4c2[2p][2(p−a)][2(p−b)][2(p−c)]
= 4
c2p(p−a)(p−b)(p−c) ❉❛í✿ h= 2
c
p
p(p−a)(p−b)(p−c)✳
❈♦♠♦ ❛ ár❡❛ S é ✐❣✉❛❧ ❛
S = 1 2ch=
c
2 2
c
p
p(p−a)(p−b)(p−c)
S=pp(p−a)(p−b)(p−c).
✶✳✷✳✹ ➪r❡❛ ❞❡ ✉♠ P♦❧í❣♦♥♦ ◗✉❛❧q✉❡r
❈♦♥❤❡❝✐❞❛s ❛s ár❡❛s ❞♦ r❡tâ♥❣✉❧♦✱ q✉❛❞r❛❞♦✱ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡ tr✐â♥❣✉❧♦✱ ♣♦❞❡♠♦s ✉t✐❧✐③á✲❧❛s ♣❛r❛ ♦❜t❡r ❛ ár❡❛ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ q✉❛❧q✉❡r✱ s✉❜❞✐✈✐❞✐♥❞♦✲♦ ❞❡ ❢♦r♠❛ ❝♦♥✈❡♥✐❡♥t❡ ❡♠ ✜❣✉r❛s ❝✉❥❛ ár❡❛ ❥á s❛❜❡♠♦s ❝❛❧❝✉❧❛r✳ ❆ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ q✉❡ s❡ q✉❡r ❡♥❝♦♥tr❛r s❡rá ❛ s♦♠❛ ❞❛s ár❡❛s ❞❛s ✜❣✉r❛s ❡♠ q✉❡ ❡st❡ ❢♦✐ s✉❜❞✐✈✐❞✐❞♦✳
❚♦♠❡♠♦s ♣r✐♠❡✐r♦ ❝♦♠♦ ❡①❡♠♣❧♦ ✉♠ tr❛♣é③✐♦ ABCD✳
❊①❡♠♣❧♦ ✶✿
❋✐❣✉r❛ ✶✳✶✷✿ P♦❧í❣♦♥♦ ❈♦♥✈❡①♦
❈♦♥s✐❞❡r❡♠♦s ❛s ❜❛s❡s AB =b1 ❡CD =b2 ❡ ❛ ❛❧t✉r❛ ❞♦ tr❛♣é③✐♦ DE =a✳ ❖ s❡❣♠❡♥t♦ ❞❡ r❡t❛AD❞✐✈✐❞❡ ♦ tr❛♣é③✐♦ ♥♦s tr✐â♥❣✉❧♦sABD❡ACD✱ ❝♦♠ ❜❛s❡s
b1 ❡ b2 r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ ♠❡s♠❛ ❛❧t✉r❛ a✳ ❆ ár❡❛ ❞♦ tr❛♣é③✐♦ é ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s ❞♦✐s tr✐â♥❣✉❧♦s✿
➪r❡❛ ❞♦ tr❛♣é③✐♦ ABCD= ab1 2 +
ab2
2 =
a(b1+b2)
2 ✳
❆❣♦r❛✱ t♦♠❡♠♦s ❝♦♠♦ s❡❣✉♥❞♦ ❡①❡♠♣❧♦ ✉♠ ❧♦s❛♥❣♦ ABCD✳
❊①❡♠♣❧♦ ✷✿
❋✐❣✉r❛ ✶✳✶✸✿ ▲♦s❛♥❣♦ ❆❇❈❉
◆♦ ❧♦s❛♥❣♦ ABCD✱ ❝♦♥s✐❞❡r❡♠♦s ❛s ❞✐❛❣♦♥❛✐s AC =d1 ❡ BD =d2✳ ◆♦t❡ q✉❡ ❛ ❞✐❛❣♦♥❛❧ AC s✉❜❞✐✈✐❞❡ ♦ ❧♦s❛♥❣♦ ❡♠ ❞♦✐s tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s✱ ABC ❡ ACD✱
❝♦♠ ❜❛s❡ ❝♦♠✉♠ AC =d1 ❡ ❛❧t✉r❛s BO ❡ DO ✐❣✉❛✐s ❛
d2
2 ✳
❉❡ss❡ ♠♦❞♦✱ ❛ ár❡❛ ❞♦ ❧♦s❛♥❣♦ s❡rá ❞❛❞❛ ♣♦r✿
➪r❡❛ ❞♦ ❧♦s❛♥❣♦ ABCD= 2
d1
d2
2 2
=
d1d2
2
❊①❡♠♣❧♦ ✸✿ ➪r❡❛ ❞❡ P♦❧í❣♦♥♦s ❘❡❣✉❧❛r❡s
❯♠ ♣♦❧í❣♦♥♦ é ❝❤❛♠❛❞♦ r❡❣✉❧❛r s❡ t♦❞♦s ♦s s❡✉s ❧❛❞♦s ❡ t♦❞♦s ♦s s❡✉s â♥❣✉❧♦s ✐♥t❡r♥♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳ P❛r❛ s❡ ♦❜t❡r ❛ ár❡❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❧✐♠✐t❛❞❛ ♣♦r ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r✱ é ♣r❡❝✐s♦ ❝♦♥s✐❞❡r❛r q✉❡ t♦❞♦ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ❞❡ ❧❛❞♦ l ✱ ♣♦❞❡
s❡r ❞✐✈✐❞✐❞♦ ❡♠ n tr✐â♥❣✉❧♦s ✐❣✉❛✐s✱ s❡♥❞♦ n ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦✳ ❈❛❞❛
tr✐â♥❣✉❧♦ t❡♠ ❝♦♠♦ ❜❛s❡ ♦ ❧❛❞♦l❞♦ ♣♦❧í❣♦♥♦ ❡ ❛❧t✉r❛ ✐❣✉❛❧ ❛♦ ❛♣ót❡♠❛ ❛ ❞♦ ♣♦❧í❣♦♥♦✳
▲❡♠❜r❛♥❞♦✿ ❆♣ót❡♠❛ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r é ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ✉♥❡ ♦ ❝❡♥tr♦ ❞❡ss❡ ♣♦❧í❣♦♥♦ ❛♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ q✉❛❧q✉❡r ✉♠ ❞❡ s❡✉s ❧❛❞♦s✳
❋✐❣✉r❛ ✶✳✶✹✿ P♦❧í❣♦♥♦s ❘❡❣✉❧❛r❡s
❉❡ss❡ ♠♦❞♦✱ ❛ ár❡❛ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ♣♦❞❡ s❡r ♦❜t✐❞❛ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ❛ ár❡❛ ❞❡ ❝❛❞❛ tr✐â♥❣✉❧♦ ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦sn ❞♦ ♣♦❧í❣♦♥♦✱ ♦ q✉❡ r❡s✉❧t❛ ❡♠✿
➪r❡❛ ❞♦ ♣♦❧í❣♦♥♦ ❂ n
l·a
2
❙❡♥❞♦ n·l = 2p ✭♣❡rí♠❡tr♦✮✱ ✈❡♠ q✉❡✿
➪r❡❛ ❞♦ ♣♦❧í❣♦♥♦ ❂ 2·p·a
2
➪r❡❛ ❞♦ ♣♦❧í❣♦♥♦ ❂ p·a✱ s❡♥❞♦ p ❂ s❡♠✐♣❡rí♠❡tr♦✳
❊①❡♠♣❧♦ ✹✿ ➪r❡❛ ❞❡ P♦❧í❣♦♥♦s ■rr❡❣✉❧❛r❡s
❯♠ ♣♦❧í❣♦♥♦ ✐rr❡❣✉❧❛r ♥ã♦ t❡♠ t♦❞♦s ♦s s❡✉s ❧❛❞♦s ✐❣✉❛✐s✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ s❡✉s â♥❣✉❧♦s ✐♥t❡r♥♦s t❛♠❜é♠ ♥ã♦ sã♦ ❝♦♥❣r✉❡♥t❡s✳ P♦❞❡♠♦s ❞❡❝♦♠♣♦r q✉❛❧q✉❡r ♣♦❧í❣♦♥♦ ✐rr❡❣✉❧❛r ❡♠ ♣♦❧í❣♦♥♦s ❞❡ ár❡❛s ❝♦♥❤❡❝✐❞❛s✳ ❖❜s❡r✈❡ ♦ ❡①❡♠♣❧♦ ❞❛ ✜❣✉r❛ ❛♦ ❛❜❛✐①♦✳ ❚❡♠♦s q✉❡ ❛ ár❡❛ ❞❡ss❡ ♣♦❧í❣♦♥♦ ✐rr❡❣✉❧❛r s❡rá ✐❣✉❛❧ ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦s tr✐â♥❣✉❧♦sT1✱T2✱ T3 ❡T4✱ ♦✉ s❡❥❛✿
Ap =T1 +T2+T3+T4.
❋✐❣✉r❛ ✶✳✶✺✿ P♦❧í❣♦♥♦ ■rr❡❣✉❧❛r
❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❞❡
❆rq✉✐♠❡❞❡s
✷✳✶ ❯♠❛ ❆❜♦r❞❛❣❡♠ ❍✐stór✐❝❛
❆rq✉✐♠❡❞❡s ✭❆r❦❤✐♠❡❞❡s✮✱ ♠❛t❡♠át✐❝♦ ❣r❡❣♦✱ ♥❛s❝✐❞♦ ❡♠ ❙✐r❛❝✉s❛ ♥❛ ❙✐❝í❧✐❛ ❡♠ ✷✽✼ ❛✳❈✳✳ ❊st✉❞♦✉✱ ❞❡s❞❡ ❥♦✈❡♠✱ ❡♠ ❆❧❡①❛♥❞r✐❛ ✭❊❣✐t♦✮ ♦♥❞❡ ❝♦♥✈✐✈❡✉ ❝♦♠ ♦s ❣r❛♥✲ ❞❡s ❣❡ô♠❡tr❛s ❞❛ é♣♦❝❛✳ ❍❛❜✐t✉❛❞♦✱ ❝♦♠ ❛s ❝♦st✉♠❡s ❞❛ s♦❝✐❡❞❛❞❡ ❛r✐st♦❝rát✐❝❛ ❡♠ q✉❡ ✈✐✈✐❛✱ ❛ ♥ã♦ ✈❛❧♦r✐③❛r ♦ tr❛❜❛❧❤♦ ♠❛♥✉❛❧✱ ♣r♦❝✉r❛♥❞♦ s❡♠♣r❡ ✉♠❛ ❥✉st✐✜❝❛t✐✈❛ ❧ó❣✐❝❛ ♣❛r❛ ❛s ❝♦♥❝❧✉sõ❡s q✉❡ ♦❜t✐♥❤❛ ❞♦s ❡♥❣❡♥❤♦s ♠❡❝â♥✐❝♦s q✉❡ ❝♦♥str✉í❛✳ ❆s ❛t✐✈✐❞❛❞❡s ❞❡ s❡✉ ♣❛✐✱ ♦ ❛strô♥♦♠♦ ❋í❞✐❛s✱ ✐♥✢✉❡♥❝✐❛r❛♠✱ s❡♠ ❞ú✈✐❞❛✱ ♥❛ ✈♦❝❛çã♦ ❡ ❢♦r♠❛çã♦ ❝✐❡♥tí✜❝❛ ❞❡ ❆rq✉✐♠❡❞❡s✳
❉♦t❛❞♦ ❞❡ ✉♠❛ ✐♥t❡❧✐❣ê♥❝✐❛ ♣r♦❞✐❣✐♦s❛✱ ❆rq✉✐♠❡❞❡s ❛ss✐♠✐❧♦✉ r❛♣✐❞❛♠❡♥t❡ t♦❞♦s ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❛❞q✉✐r✐❞♦s ♣❡❧❛ ❤✉♠❛♥✐❞❛❞❡ ❛té ❛q✉❡❧❡ ♠♦♠❡♥t♦ ❡✱ ❛tr❛✈és ❞❡ ✉♠❛ ❛❞♠✐rá✈❡❧ sér✐❡ ❞❡ ❞❡s❝♦❜❡rt❛s✱ ❛♠♣❧✐♦✉✲♦s ❣r❛♥❞❡♠❡♥t❡✳
❆rq✉✐♠❡❞❡s ♣r♦❡♠✐♥❡♥t❡ ♠❛t❡♠át✐❝♦ ❡ ✐♥✈❡♥t♦r ❣r❡❣♦✱ ❡s❝r❡✈❡✉ ✐♠♣♦rt❛♥t❡s tr❛✲ ❜❛❧❤♦s s♦❜r❡ ❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡ só❧✐❞❛✱ ❛r✐t♠ét✐❝❛ ❡ ♠❡❝â♥✐❝❛✳ ❙❡♠ ❞ú✈✐❞❛ ♦ ♠❛✐♦r ❣ê♥✐♦ ❞❛ ❆♥t✐❣ü✐❞❛❞❡ ❝❧áss✐❝❛ ❡ ✉♠ ❞♦s ♠❛✐♦r❡s ❞❡ t♦❞♦s ♦s t❡♠♣♦s✱ ❆rq✉✐♠❡❞❡s r❡ú♥❡ t♦❞❛s ❛s ❝❛r❛❝t❡ríst✐❝❛s q✉❡ ♦ ✐♠❛❣✐♥ár✐♦ ♣♦♣✉❧❛r ❛tr✐❜✉✐ ❛ ✉♠ ✈❡r❞❛❞❡✐r♦ sá❜✐♦✳
❙❡✉s ♠ét♦❞♦s ❛♥t❡❝✐♣❛r❛♠ ♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧✱ ✷✵✵✵ ❛♥♦s ❛♥t❡s ❞❡ t❡r s✐❞♦ ✧❝r✐✲ ❛❞♦✧♣♦r ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✳ ❆rq✉✐♠❡❞❡s t❛♠❜é♠ ♣r♦✈♦✉ q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ❝♦rr❡s♣♦♥❞❡ ❛ ❞♦✐s t❡rç♦s ❞♦ ✈♦❧✉♠❡ ❞♦ ❝✐❧✐♥❞r♦ ❝✐r❝✉♥s❝r✐t♦✳ ❊✈✐❞❡♥t❡♠❡♥t❡ ❡❧❡ ❝♦♥s✐❞❡r♦✉ ❡st❡ ❝♦♠♦ s❡✉ ♠❛✐♦r ❢❡✐t♦✱ ♣♦✐s ♣❡❞✐✉ q✉❡ s✉❛ ❧á♣✐❞❡ t✐✈❡ss❡ ✉♠❛ ❡s❢❡r❛
❝✐r❝✉♥s❝r✐t❛ ♣♦r ✉♠ ❝✐❧✐♥❞r♦✳
❊♠ ●❡♦♠❡tr✐❛✱ ♦ sá❜✐♦ t❡✈❡ ♦ ♠ér✐t♦ ❞❡ ❝♦♥❝❡❜❡r ♠ét♦❞♦s ❣❡r❛✐s ♣❛r❛ ❝❛❧❝✉❧❛r ❛s ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❝✉r✈✐❧í♥❡❛s ❡ ♦s ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s ❞❡❧✐♠✐t❛❞♦s ♣♦r s✉♣❡r❢í✲ ❝✐❡s ❝✉r✈❛s✳ ❆♣❧✐❝♦✉ t❛✐s s✐st❡♠❛s ❛ ✈ár✐♦s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s✿ à ❡s❢❡r❛✱ ❛♦ ❝ír❝✉❧♦✱ ❛♦ s❡❣♠❡♥t♦ ❞❡ ♣❛rá❜♦❧❛✱ à ár❡❛ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❞♦✐s r❛✐♦s ❡ ❞♦✐s ♣❛ss♦s s✉❝❡ss✐✈♦s ❞❡ ✉♠❛ ❡s♣✐r❛❧✱ ❛♦s s❡❣♠❡♥t♦s ❡s❢ér✐❝♦s✱ às s✉♣❡r❢í❝✐❡s ❣❡r❛❞❛s ♣❡❧❛s r❡✈♦❧✉çõ❡s ❡♠ t♦r♥♦ ❞♦s ❡✐①♦s ♣r✐♥❝✐♣❛✐s ❞♦s r❡tâ♥❣✉❧♦s ✭♦✉ ♠❡❧❤♦r✱ ♦s ❝✐❧✐♥❞r♦s✮✱ ❛ ❡♥t✐❞❛❞❡s ❣❡♦✲ ♠étr✐❝❛s ♣r♦❞✉③✐❞❛s ♣❡❧❛ r❡✈♦❧✉çã♦ ❞♦s tr✐â♥❣✉❧♦s ✭♦✉ s❡❥❛✱ ♦s ❝♦♥❡s✮✱ ❞❛s ♣❛rá❜♦❧❛s ✭♣❛r❛❜♦❧ó✐❞❡s✮✱ ❞❛s ❤✐♣ér❜♦❧❡s ✭❤✐♣❡r❜♦❧ó✐❞❡s✮ ❡ ❞❛s ❡❧✐♣s❡s ✭❡❧✐♣só✐❞❡s✮✳ ❆rq✉✐♠❡✲ ❞❡s t✐♥❤❛✱ ♣♦rt❛♥t♦✱ ✉♠ s✐st❡♠❛ ❞❡ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧ ❞♦✐s ♠✐❧ ❛♥♦s ❛♥t❡s ❞❡ ◆❡✇t♦♥ ❡ ▲❡✐❜♥✐③✳
❆rq✉✐♠❡❞❡s ♥ã♦ ❛♥t❡❝✐♣❛ ❛♣❡♥❛s ♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧✱ ♥❛ ✈❡r❞❛❞❡✱ ✉♠❛ ❞❛s s✉❛s ♠❛✐s ❝♦♥❤❡❝✐❞❛s ❡ ✐♠♣♦rt❛♥t❡s ❞❡s❝♦❜❡rt❛s ♠❛t❡♠át✐❝❛s é ❛ ❝♦♥str✉çã♦ ❞❛ ❢❛♠♦s❛ ❡s♣✐r❛❧ ❞❡ ❆rq✉✐♠❡❞❡s✳
❆❧é♠ ❞✐ss♦✱ ❢❡③ s✉r❣✐r ❛ ✐❞❡✐❛ ❞❡ ✐♥✜♥✐t❛♠❡♥t❡ ❣r❛♥❞❡ ❛♦ q✉❡r❡r ❝♦♥t❛r ♦s ❣rã♦s ❞❡ ❛r❡✐❛ ❞❛ ♣r❛✐❛ ❞❡ ❙✐r❛❝✉s❛✳ ❊st❛ ❛❜♦r❞❛❣❡♠ ❞❛ ✐❞❡✐❛ ❞❡ ✐♥✜♥✐t♦ s✉r❣❡ t❛♠❜é♠ ♥✉♠❛ ❞❛s s✉❛s ♦❜r❛s✱ ♦♥❞❡ s❡ ♣r♦♣õ❡ ❛✈❛❧✐❛r ♦ ♥ú♠❡r♦ ❞❡ ❣rã♦s ❞❡ ❛r❡✐❛ q✉❡ s❡r✐❛ ♣r❡❝✐s♦ ♣❛r❛ ❡♥❝❤❡r ✉♠❛ ❡s❢❡r❛ ❣r❛♥❞❡ ❝♦♠♦ ♦ ❯♥✐✈❡rs♦✳ P❛r❛ r❡s♦❧✈❡r ❡st❡ ♣r♦❜❧❡♠❛✱ t❡✈❡ ❞❡ ✉❧tr❛♣❛ss❛r ❞✉❛s ❞✐✜❝✉❧❞❛❞❡s✿ ❛ ♣r✐♠❡✐r❛✱ ❞❛r ❛s ❞✐♠❡♥sõ❡s ❞♦ ✉♥✐✈❡rs♦❀ ❛ s❡❣✉♥❞❛✱ ❝r✐❛r ✉♠ ♠♦❞♦ ❞❡ ❡s❝r❡✈❡r ♦ ♥ú♠❡r♦ ❝♦❧♦ss❛❧ ❞♦s ❣rã♦s ❞❡ ❛r❡✐❛✳ ❚❛r❡❢❛ t❛♥t♦ ♠❛✐s ❞✐❢í❝✐❧ q✉❛♥t♦ à ❡s❝r✐t❛ ❣r❡❣❛ ❞♦s ♥ú♠❡r♦s só ♣❡r♠✐t✐❛ ❡s❝r❡✈❡r ♥ú♠❡r♦s ✐♥❢❡r✐♦r❡s à ♠✐rí❛❞❡ ❞❛s ♠✐rí❛❞❡s ✭✶✵✵✳✵✵✵✳✵✵✵✮✳
✷✳✷ ❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦
❆rq✉✐♠❡❞❡s t❛♠❜é♠ ❛♣r❡s❡♥t♦✉ ✉♠❛ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❛ ♣❛r✲ t✐r ❞♦ ♠ét♦❞♦ ❞❛ ❡①❛✉stã♦✱ t❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ♣♦r Pr✐♥❝í♣✐♦ ❞❡ ❊✉❞♦①♦✲❆rq✉✐♠❡❞❡s✱ ♣♦✐s t❡♠ ❝♦♠♦ ❜❛s❡ ❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❛♣r❡s❡♥t❛❞❛ ♣♦r ❊✉❞♦①♦ ❞❡ ❈♥✐❞♦ ✭✹✵✽ ❛✳❈✳ ✲ ✸✺✺ ❛✳❈✳✮ ❡ ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛ ✭✷✽✼ ❛✳❈✳ ✲ ✷✶✷ ❛✳❈✳✮✱ q✉❡ ❢♦✐ ♦ ♠❛t❡♠át✐❝♦ q✉❡ ♠❛✐s ❡①♣❧♦r♦✉ ❡ss❡ ♠ét♦❞♦ ♥❛ ❛♥t✐❣✉✐❞❛❞❡ ✭P■◆❚❖✱ ✷✵✵✹✮✳
❊✉❞♦①♦ ❞❡✜♥❡ ✉♠❛ t❡♦r✐❛ ❞❡ ♣r♦♣♦rçõ❡s q✉❡ é ❛♣❧✐❝á✈❡❧ t❛♥t♦ ❡♠ ❣r❛♥❞❡③❛s ♠❡♥s✉rá✈❡✐s q✉❛♥t♦ ❡♠ ❣r❛♥❞❡③❛s ✐♥❝♦♠❡♥s✉rá✈❡✐s✱ t♦r♥❛♥❞♦ ✉❧tr❛♣❛ss❛❞❛ ❛ t❡♦r✐❛ ❛r✐t♠ét✐❝❛ ❞♦s ♣✐t❛❣ór✐❝♦s✳
P❛r❛ ❝❤❡❣❛r♠♦s ♥❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❞❡ ❊✉❞♦①♦✱ ✈❛♠♦s ♦❜s❡r✈❛r ❛❧❣✉♠❛s
❞❡✜♥✐çõ❡s ❞❡ ❊✉❝❧✐❞❡s✿
❉❡✜♥✐çã♦ ✸ ❞♦ ▲✐✈r♦ ❱✿ ❯♠❛ r❛③ã♦ é ✉♠❛ ❡s♣é❝✐❡ ❞❡ r❡❧❛çã♦ ❛ r❡s♣❡✐t♦ ❞♦ t❛♠❛✲ ♥❤♦ ❡♥tr❡ ❞✉❛s ❣r❛♥❞❡③❛s ❞♦ ♠❡s♠♦ t✐♣♦✳
❉❡✜♥✐çã♦ ✹ ❞♦ ▲✐✈r♦ ❱✿ ❉✐③✲s❡ q✉❡ t❡♠ ✉♠❛ r❛③ã♦ ❛s ❣r❛♥❞❡③❛s q✉❡ sã♦ ❝❛♣❛③❡s✱ q✉❛♥❞♦ ♠✉❧t✐♣❧✐❝❛❞❛s✱ ❞❡ s❡ ❡①❝❡❞❡r ✉♠❛ à ♦✉tr❛✳
❉❡✜♥✐çã♦ ✺ ❞♦ ▲✐✈r♦ ❱✿ ❉✐③✲s❡ q✉❡ ❣r❛♥❞❡③❛s ❡stã♦ ♥❛ ♠❡s♠❛ r❛③ã♦✱ ❛ ♣r✐♠❡✐r❛ ♣❛r❛ ❛ s❡❣✉♥❞❛ ❡ ❛ t❡r❝❡✐r❛ ♣❛r❛ q✉❛rt❛✱ q✉❛♥❞♦✱ ❞❛❞♦s q✉❛✐sq✉❡r ❡q✉✐♠ú❧t✐♣❧♦s ❞❛ ♣r✐♠❡✐r❛ ❡ ❞❛ t❡r❝❡✐r❛ ❡ ❞❛❞♦s q✉❛✐sq✉❡r ❡q✉✐♠ú❧t✐♣❧♦s ❞❛ s❡❣✉♥❞❛ ❡ ❞❛ q✉❛rt❛✱ ♦s ♣r✐♠❡✐r♦s ❡q✉✐♠ú❧t✐♣❧♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❡①❝❡❞❡♠✱ sã♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ✐❣✉❛✐s ♦✉ ✜❝❛♠ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❛q✉é♠ ❞♦s ú❧t✐♠♦s✳ ❊st❛ ❞❡✜♥✐çã♦ é ❝♦♥s♦❧✐❞❛❞❛ ♥❛ ❉❡✜♥✐çã♦ ✻✱ ❞♦ ♠❡s♠♦ ▲✐✈r♦✿ ●r❛♥❞❡③❛s q✉❡ tê♠ ❛ ♠❡s♠❛ r❛③ã♦ ❞✐③❡♠✲s❡ ♣r♦♣♦r❝✐♦♥❛✐s✳
❆ ♥♦t❛çã♦ ❛t✉❛❧✱ q✉❡ tr❛❞✉③ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❊✉❝❧✐❞❡s✱ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ a
b = c
d s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❞❛❞♦s ♦s ✐♥t❡✐r♦s m ❡n✱ s❡♠♣r❡ q✉❡ ma < nb✱ ❡♥tã♦ mc < nd❀ ♦✉ s❡ ma=nb✱ ❡♥tã♦ mc=nd❀ ♦✉ s❡ ma > nb✱ ❡♥tã♦ mc > nd✳ ◆♦t❡✲s❡
q✉❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ❊✉❞♦①♦ ❞❡ ✐❣✉❛❧❞❛❞❡ ❞❡ r❛③õ❡s ❝♦♥❞✉③✲♥♦s ❛♦ ♣r♦❝❡ss♦ ❞❡ r❡❞✉çã♦ ❛♦ ♠❡s♠♦ ❞❡♥♦♠✐♥❛❞♦r✱ ♦✉ s❡❥❛✿ ad = bc✱ q✉❡ ♥ã♦ é ♠❛✐s ❞♦ q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦
❝r✉③❛❞❛✱ ✉s❛❞❛ ❤♦❥❡ ♥❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ❢r❛çõ❡s✱ ♦ q✉❡ ❢♦r♠❛❧♠❡♥t❡ ♥ã♦ ❡r❛ ❢❡✐t♦ ♣❡❧♦s ❣r❡❣♦s à é♣♦❝❛ ❞❡ ❊✉❝❧✐❞❡s ✭❇♦②❡r✱ ✶✾✾✻✮✳
❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❧ó❣✐❝♦✱ ❡st❛s ❞✉❛s ❞❡✜♥✐çõ❡s r❡❞✉③❡♠ ❛ ♥♦çã♦ ❞❡ ♣r♦♣♦rçã♦ ❡♥tr❡ ❞♦✐s ♣❛r❡s ❞❡ ❣r❛♥❞❡③❛s ❤♦♠♦❣ê♥❡❛s à ♥♦çã♦ ❞❡ ♦r❞❡♠ ❡♥tr❡ ♠ú❧t✐♣❧♦s ❞❡ss❛s ❣r❛♥❞❡③❛s ✭❙á✱ ✷✵✵✵✮✳
❉❡✜♥✐çã♦ ✼ ❞♦ ▲✐✈r♦ ❱✿ ◗✉❛♥❞♦✱ ❞♦s ❡q✉✐♠ú❧t✐♣❧♦s✱ ♦ ♠ú❧t✐♣❧♦ ❞❛ ♣r✐♠❡✐r❛ ❣r❛♥✲ ❞❡③❛ ❡①❝❡❞❡ ♦ ♠ú❧t✐♣❧♦ ❞❛ s❡❣✉♥❞❛✱ ♠❛s ♦ ♠ú❧t✐♣❧♦ ❞❛ t❡r❝❡✐r❛ ♥ã♦ ❡①❝❡❞❡ ♦ ♠ú❧t✐♣❧♦ ❞❛ q✉❛rt❛✱ ❞✐③✲s❡ q✉❡ ❛ ♣r✐♠❡✐r❛ t❡♠ ✉♠❛ r❛③ã♦ ♠❛✐♦r ♣❛r❛ s❡❣✉♥❞❛ ❞♦ q✉❡ ❛ t❡r❝❡✐r❛ ♣❛r❛ q✉❛rt❛✳ ❊st❛ ❞❡✜♥✐çã♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ♣❛r❛ q✉❛✐sq✉❡r ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐sm❡ n✱ q✉❛♥❞♦ ❢♦r ✈❡r❞❛❞❡✐r❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ma > nb✱ ❡ q✉❛♥❞♦ ❢♦r ❢❛❧s❛ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ mc > nd ✱ ❡♥tã♦ ❞✐③✲s❡ q✉❡ a
b > c
d ✭❙á✱ ✷✵✵✸✮✳
P❛r❛ ✜♥❛❧✐③❛r ❡ss❛ ❜r❡✈❡ ♣❛ss❛❣❡♠ ♣❡❧❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ❞❡ ❊✉①♦❞♦✱ ❞❡✈❡✲ ♠♦s ❢❛❧❛r ♥❛ ❞❡♠♦♥str❛çã♦✱ ♥♦ ❝❛s♦ ❞❡ tr✐â♥❣✉❧♦s✱ ❞❛ ♣r♦♣♦s✐çã♦ ✶ ❞♦ ▲✐✈r♦ ❱■ ❞❡ ❊✉❝❧✐❞❡s✿ ❚r✐â♥❣✉❧♦s ❡ ♣❛r❛❧❡❧♦❣r❛♠♦s s♦❜ ❛ ♠❡s♠❛ ❛❧t✉r❛ ❡stã♦ ❡♥tr❡ s✐ ❝♦♠♦ ❛s s✉❛s ❜❛s❡s✱ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❙á ✭✷✵✵✵✮✱ ✉s❛♥❞♦ ♦s ❡q✉✐♠ú❧t✐♣❧♦s✱ ❝♦♥t♦r♥❛♥❞♦ ❞❡st❡ ♠♦❞♦ ❛ ✐♥❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡ q✉❡ ❢❡③ ❝♦♠ q✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❛♣r❡s❡♥t❛❞❛ ♣❡❧♦s ♣✐t❛❣ór✐❝♦s ❞❡✐①❛ss❡ ❞❡ s❡r ❛❝❡✐t❛✳
❊❧❡♠❡♥t♦s ❳✱ ✶✿ ❉❛❞❛s ❞✉❛s ❣r❛♥❞❡③❛s ❞❡s✐❣✉❛✐s✱ s❡ ❞❛ ♠❛✐♦r s✉❜tr❛✐r✲s❡ ✉♠❛ ❣r❛♥❞❡③❛ ♠❛✐♦r ❞♦ q✉❡ ❛ s✉❛ ♠❡t❛❞❡✱ ❡ ❞♦ q✉❡ s♦❜r❛r ✉♠❛ ❣r❛♥❞❡③❛ ♠❛✐♦r ❞♦ q✉❡
❛ s✉❛ ♠❡t❛❞❡✱ ❡ s❡ ❡st❡ ♣r♦❝❡ss♦ ❢♦r r❡♣❡t✐❞♦ ❝♦♥t✐♥✉❛♠❡♥t❡✱ s♦❜r❛rá ✉♠❛ ❣r❛♥❞❡③❛ ♠❡♥♦r ❞♦ q✉❡ ❛ ♠❡♥♦r ❞❛s ❣r❛♥❞❡③❛s ❞❛❞❛s✳
❉❡ ❢❛t♦✳ ❈♦♥s✐❞❡r❡♠♦s a❡b ❞✉❛s ❣r❛♥❞❡③❛s ❞♦ ♠❡s♠♦ t✐♣♦ ✭✷✳✶❛✮ ❡ s✉♣♦♥❤❛✲s❡✱
s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ a > b✳ ❆t❡♥❞❡♥❞♦ à ❞❡✜♥✐çã♦ ✹ ❞❡ ❊❧❡♠❡♥t♦s ❱✱
❡①✐st❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n✱ t❛❧ q✉❡ nb > a✳
✭❛✮ ❛❃❜ ✭❜✮ ❜❃♥❛
❋✐❣✉r❛ ✷✳✶✿ ❙❡❣♠❡♥t♦s ❞❡ r❡t❛
◆❡st❛s ❝♦♥❞✐çõ❡s✱ t♦♠❡♠♦s ❛s ❣r❛♥❞❡③❛sa❡nb✭✷✳✶❜✮✳ ❙❡ ❛ar❡t✐r❛r♠♦s ♠❛✐s ❞❛
♠❡t❛❞❡✱ ❡ ❛nb r❡t✐r❛r♠♦s b ✭q✉❡ é ♠❡♥♦s ❞♦ q✉❡ ❛ ♠❡t❛❞❡ ❞❡nb✮✱ r❡st❛♠✲♥♦s ❞✉❛s
❣r❛♥❞❡③❛s✿ a1 <
1
2a ❡ (n−1)b✱ t❛✐s q✉❡ (n−1)b > a1 ✭✷✳✷❛✮✳ ❙❡✱ ♣♦r ✉♠ ♣r♦❝❡ss♦
✐❞ê♥t✐❝♦ ❛♦ ❛♥t❡r✐♦r✱ ❛ a1 s❡ r❡t✐r❛r ♠❛✐s ❞❛ ♠❡t❛❞❡ ❡ ❛ (n−1)b r❡t✐r❛r ♥♦✈❛♠❡♥t❡
b ✭q✉❡ é ♠❡♥♦s ❞♦ q✉❡ ♠❡t❛❞❡ ❞❡ (n−1)b✮ ✜❝❛r❡♠♦s ❝♦♠ ❞✉❛s ❣r❛♥❞❡③❛s✿ a2 < a1 ❡ (n−2)b✱ t❛✐s q✉❡ (n−2)b > a2✳
✭❛✮(n−1)b > a1 ✭❜✮ ♥✲✷ ♣❛ss♦s
❋✐❣✉r❛ ✷✳✷✿ ❉❡♠♦♥str❛çã♦ ❞♦ ▼ét♦❞♦ ❞❛ ❊①❛✉stã♦
❆♦ ✜♠ ❞❡ (n−2) ♣❛ss♦s✱ ♦❜t❡♠♦s ✉♠❛ ❣r❛♥❞❡③❛ an−2 t❛❧ q✉❡ 2b > an−2✳ ❙❡ ❛
an−2 ♠❛✐s ❞❛ ♠❡t❛❞❡✱ ❡ ❛ 2b r❡t✐r❛r b✱ s♦❜r❛ ✉♠❛ ❣r❛♥❞❡③❛ an−1 t❛❧ q✉❡ b > an−1 ✭♣♦✐s ❛2b r❡t✐r♦✉✲s❡ ❡①❛t❛♠❡♥t❡ ❛ ♠❡t❛❞❡✮✳ ❆ss✐♠✱ ❛♦ ✜♠ ❞❡(n−1)♣❛ss♦s✱ ♦❜té♠✲
s❡ ✉♠❛ ❣r❛♥❞❡③❛ an−1 ♠❡♥♦r ❞♦ q✉❡ b✱ ❛ ♠❡♥♦r ❞❛s ❣r❛♥❞❡③❛s ✐♥✐❝✐❛❧♠❡♥t❡ ❞❛❞❛s ✭✷✳✷❜✮✱ ♦ q✉❡ ♣r♦✈❛ ♦ ♠ét♦❞♦ ❞❛ ❡①❛✉stã♦ ✭♣r✐♥❝í♣✐♦ ❞❡ ❊✉❞♦①♦✲❆rq✉✐♠❡❞❡s✮✳
◆♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♣♦❞❡✲s❡ ❞✐③❡r q✉❡ ♦ ♠ét♦❞♦ ❞❛ ❡①❛✉stã♦ ❝♦♥s✐st❡ ❡♠ ❜✉s❝❛r
❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s ❞❛ ár❡❛ ❛ s❡r ♠❡❞✐❞❛✱ ♣♦r ❢❛❧t❛ ❡ ♣♦r ❡①❝❡ss♦✱ ❛ ♣❛rt✐r ❞❡ ♦✉tr❛s ❥á ❝♦♥❤❡❝✐❞❛s✳
❈❤❛♠❛r❡♠♦s ❞❡ S ❛ ár❡❛ ❞❛ ✜❣✉r❛ ❛ s❡❣✉✐r✱ ❝✉❥❛ ár❡❛ s❡ ❞❡s❡❥❛ ♦❜t❡r✳
❋✐❣✉r❛ ✷✳✸✿ ❋✐❣✉r❛ ❞❡ ár❡❛S
❈❤❛♠❛r❡♠♦s ❞❡ A ❛ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦ ✐♥t❡r♥♦ ❛ ❙✱ ❡ ❞❡ ❇ ❛ ár❡❛ ❞♦ ♣♦❧í❣♦♥♦
❡①t❡r♥♦ ❛ ❙✳
❋✐❣✉r❛ ✷✳✹✿ P♦❧í❣♦♥♦ ✐♥t❡r✐♦r ❞❡ ár❡❛ ❆
P♦❞❡♠♦s ❞✐③❡r q✉❡ ❛ ár❡❛ A é ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ❡B é ❛ ❛♣r♦①✐♠❛çã♦ ♣♦r
❡①❝❡ss♦ ❞❛ ár❡❛ S✳ ❖✉ s❡❥❛✿ A≤S ≤B.
✷✳✷✳✶ ❖ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ❡ ❛ ➪r❡❛ ❞♦ ❈ír❝✉❧♦
❊❧❡♠❡♥t♦s ❳■■✱ ✷ ✿ ❈ír❝✉❧♦s ❡stã♦ ❡♥tr❡ s✐ ❝♦♠♦ ♦s q✉❛❞r❛❞♦s s♦❜r❡ ♦s ❞✐â♠❡tr♦s✳ P♦❞❡♠♦s r❡❡s❝r❡✈❡r ❊❧❡♠❡♥t♦s ❳■■✱ ✷ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ ✿ ❆ r❛③ã♦ ❡♥tr❡ ❛s ár❡❛s ❞❡ ❞♦✐s ❝ír❝✉❧♦s é ✐❣✉❛❧ à r❛③ã♦ ❡♥tr❡ ❛s ár❡❛s ❞❡ ❞♦✐s q✉❛❞r❛❞♦s ❝✉❥♦s ❧❛❞♦s sã♦ ♦s ❞✐â♠❡tr♦s ❞♦s ❝ír❝✉❧♦s✳
❈♦♥s✐❞❡r❡♠♦s ❞✉❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s ❞❡ ár❡❛s A ❡ a ✱ ❡ ❞✐â♠❡tr♦sD ❡ d✱ r❡s♣❡❝t✐✲
✈❛♠❡♥t❡✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ❛❜❛✐①♦✳
◆❡st❛s ❝♦♥❞✐çõ❡s✱ ❛ ♣r♦♣♦rçã♦ ♥♦s ❞✐③ q✉❡✿ A
a = D2
d2 ✳
❙❡❥❛♠ R ❡ r✱ t❛✐s q✉❡ D = 2R ❡ d = 2r✳ ❆ss✐♠✱ ❛s ár❡❛s ❞❛s ❝✐r❝✉♥❢❡rê♥❝✐❛s
s❡rã♦ ❞❛❞❛s ♣♦r A =πR2 ❡ a =πr2✱ ❡ ❛s ár❡❛s ❞♦s q✉❛❞r❛❞♦s s❡rã♦ D2 = (2R)2 ❡
d2 = (2r)2✳
❙❡❣✉❡✲s❡ q✉❡✿ A
a = D2
d2 ⇐⇒
πR2
πr2 =
(2R)2
(2r)2 ⇐⇒
R2
r2 =
R2
r2 ♦❦ ✦
